Decoherence theory: System+Apparatus+Environment correlated. What becomes the global state when I measure SA? Did it collapse? In decoherence theory, the basic situation is the following (I illustrate with two level system for simplicity). I want to measure a system $S$ by the mean of an apparatus $A$. Around it there is the environment $E$.
I assume the system is initially in $|\psi_S \rangle = a |0 \rangle + b |1\rangle$, the apparatus and environment in a state $|0\rangle$
The first step if the pre-measurement: system and apparatus get quantum correlated:
$$|\psi_S \rangle |0\rangle |0\rangle \rightarrow \left( a|00\rangle + b|11\rangle \right) |0\rangle $$
The second step is the correlation with the environment:
$$\left( a|00\rangle + b|11\rangle \right) |0\rangle \rightarrow a|000\rangle + b|111\rangle$$
Tracing out the environment, we find the good mixed density matrix for system-apparatus:
$$\rho_{SA}=|a|^2 |00\rangle \langle 00| + |b|^2 |11 \rangle \langle 11 |$$
My question
In decoherence theory, as far as I understood there is no need for a collapse: all the measurement can be done based on unitary dynamics as I showed. However what disturbs me is that in practice, the experimentalist will either find $|00\rangle$ or $|11\rangle$ (with probabilities $|a|^2$ or $|b|^2$).
But if there were no collapse: what does the final global state becomes ? Is it still in $a|000\rangle + b|111\rangle$ ? I would find it weird because we now know that $SA$ is in $|00\rangle$.
Then is it now in $|000\rangle$ ? If it is the case it means a collapse occured on the system $SAE$.
I am confused.

About the bounty: In the comment it appears that indeed decoherence theory doesn't say anything about when I will have read the outcome (if I find $|00\rangle$, what will the environment state be).
However what confuses me is that taking a system composed of three photons, we know for sure that if the two first are in $|00\rangle$ the last one will be in $|0\rangle$ if the initial state was $a |000\rangle+b|111\rangle$. The entanglement will be completly broken basically. And it seems to be a missing explanation from decoherence.
What I want to be 100% sure about is that indeed decoherence doesn't explain this. I would like to see a clear source in which this specific issue is stated, and where it is said it does'nt solve it. In many sources we can see that it doesn't solve the measurement problem but it is very vague. If it means that it doesn't explain the origin of probability only it doesn't mean it would'nt solve this problem.
 A: Let us denote the System + Apparatus' Hilbert space as $\mathcal{H_{SA}}$ and the Environment's Hilbert space as $\mathcal{H_E}$. Your SAE state is an entangled state which lives in the Hilbert space $\mathcal{H_{SA}} \otimes \mathcal{H_E}$. The experimentalist performs her operations on SA after tracing out the environment, where she now works with an effective theory of SA whose Hilbert space is just $\mathcal{H_{SA}}$, while the Hilbert space $\mathcal{H_E}$ ceases to exist in her description. This means that she has destroyed any entanglement between SA and E, and is left with your reduced density matrix on SA. As a result, she cannot conclude that the environment is the state $| 0\rangle$ anymore because the operation of tracing over has now reduced her Hilbert space from $\mathcal{H_{SA}} \otimes \mathcal{H_E}$ to $\mathcal{H_{SA}}$. The notion of a "global state" living on $\mathcal{H_{SA}} \otimes \mathcal{H_E}$ does not exist anymore. The experimentalist can now perform a measurement on SA and find $| 00\rangle$ or $| 11\rangle$. The measurement of the environment can now give a totally uncorrelated value as indicated from the initially entangled state, depending on its reduced density matrix.
Another way to see this is from the path integral perspective, where the transitional amplitudes are now given by "influence functional". Here you have integrated out the environment in SAE path integral, and thus are left with "influence phases" in the integral due to the SA-E interaction. Finally you are only concerned with amplitudes between in and out states living in $\mathcal{H_{SA}}$.
See Section 4  of this review, where the author talks about a similar SAE state as in the question. The preceding sections of this paper might be useful too, as well as the references therein. For the influence functional, see this pioneering paper.
